{
  "abstract": "We study decompositions of holomorphic isometries of complex hyperbolic space into holomorphic involutions. We prove that, for every n>=3, the involution length of PU(n,1) is 4. This improves the higher-dimensional upper bound 8 of Paupert--Will, as well as the projective upper bound 5 implied by Bünger's linear decomposition theorem, to the optimal value 4. Combined with the two-dimensional result of Paupert--Will, this shows that the involution length of PU(n,1) is 4 for every n>=2. The core of the proof is a three-involution decomposition theorem for square roots: every g in PU(n,1) has a square root of involution length at most 3, and this uniform bound is optimal. The key construction is carried out first on a two- or three-dimensional indefinite block and then completed in higher dimensions by a necessary and sufficient spectral pairing criterion on the positive-definite orthogonal complement. The four-factor lower bound is provided by complex reflections in a point. As a consequence, every element of PU(n,1) is a single commutator, and the two elements in the commutator representation can be simultaneously reversed by the same nontrivial holomorphic involution. This extends the two-dimensional single-commutator result of Paupert--Will to every n>=2 and thereby verifies Djoković's Conjecture A for the family PU(n,1), n>=2.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17957",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhongqi Wang"
    },
    {
      "name": "Shihai Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bj3refa5twcmj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17957",
  "fulltext_url": null,
  "id": "bj3refa5twcmj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17957",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ]
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  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.17957",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bj3refa5twcmj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17957",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17957"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Involution and Commutator Length in PU(n,1)",
  "title_text": "Involution and Commutator Length in PU(n,1)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/bj3refa5twcmj",
  "venue": "arXiv",
  "versions": []
}